Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 6 Math Circles
Rates and Ratios
November 15, 2012
What are Rates and Ratios?
Definition. A ratio is a COMPARISON between two things
(e.g. numbers, people, fruit, animals). They are written as
(how much of one thing) : (how much of the other thing)
A RATE is a special kind of ratio.
It measures how
much one quantity or thing changes with respect to another.
How to use them?
Ratios are used to express proportions - how much we have of something relative
to something else.
Example
Walter is mixing some chemicals for a science experiment. According to
the recipe, the solution needs 5 parts of hydrofluoric acid for every 2 parts
of nitric acid. He has 24 parts of nitric acid... How much hydrofluoric acid
does he need?
Example
Walter is mixing some chemicals for a science experiment. According to the recipe,
the solution needs 5 parts of hydrofluoric acid for every 2 parts of nitric acid. He has
24 parts of nitric acid... How much hydrofluoric acid does he need?
Solution
To approach this problem, we are going to find out the UNIT COST
Unit Cost Method
The key to this method is to use DIVISION.
Write the ratio describing the situation. The left number represents the amount to
be solved for: 5 : 2
Divide the left number by the right number: 5 ÷ 2 = 2.5.
The answer you just obtained is your unit cost. The unit cost tells you how many
parts hydrofluoric acid you need for every 1 part of nitric acid.
Multiply the unit cost by the actual amount of nitric acid:
2.5 × 24 = 60.
This is your final answer!
Therefore, Walter needs 60 parts hydrofluoric acid.
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Another example...
In a survey of music taste, the ratio of Pink Floyd fans to Justin Bieber fans was 4 :
16 (this is not a realistic ratio, I hope). If there were 256 Justin Bieber fans in total,
how many Pink Floyd fans were there?
Set up ratio: 4 : 16
Divide to get unit cost: 4 ÷ 16 = 0.25
Multiply unit cost by how many J. B. fans in total: 0.25 × 256 = 64
Conclude: Therefore there are 64 Pink Floyd fans in total.
Exercises - Solve these problems using the unit ratio method on another sheet of
paper.
1. Walter is back. For his new experiment, he needs 2 parts of water for every 8
parts of salt. If he has 7.5 parts of water, how many parts of salt did he use?
8 : 2 → 8 ÷ 2 = 4 → 4 × 7.5 = 30 Therefore he used 30 parts of salt.
2. In another survey of music taste, the ratio of Carly Rae Jepsen fans to Miley
Cyrus fans was found to be 5 : 8. If there were 176 Miley Cyrus fans, how many
Carly Rae fans were there?
5 : 8 → 5 ÷ 8 = 0.625 → 0.625 × 176 = 110 Therefore there were 110 Carly Rae
fans.
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Now that you’ve learned these two methods for ratios, rates will be a piece of cake!
The only difference is that rates use PHYSICAL UNITS of MEASUREMENT (km, L
and minutes are physical units of measurement; people, fruits are not. When money
is involved, we count it is a rate as well).
Example
Dexter is sailing on his boat in the middle of the ocean. He notices
a small hole in the hull, leaking water. He immediately starts back
to shore. For every 2 km, 15 L of water seep in. If Dexter is 15
km from shore, how much water will he take on when he reaches
shore?
Solution
We know that 15 L of water comes in for every 2 km. We are trying to solve for the
amount of water after 15 km.
When you set up your ratio, don’t forget units: 15 L : 2 km
Divide left number by right: 15 ÷ 2 = 7.5
This tells you how many L of water is taken on per 1 km.
To find how much water he takes on over 15 km, we multiply the unit cost by 15,
since he travels 15 km.
Therefore, he will take on 7.5 × 15 = 112.5 L by the time he reaches shore.
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One more example...
1. Johnny earns $36 for every 3 hours he works in his dad’s shop. If he works 45
hours a week, how much does he earn in a week?
$36 : 3 hours → 36 ÷ 3 = 12 → 12 × 45 = 540, since he works 45 hours in total.
Therefore he makes $540 a week.
Now you try... (on another sheet of paper)
1. From another of our examples, if the pipes in the basement leak 3 litres of water
every 2 weeks, how much is lost over 9 weeks?
3 : 2 → 3 ÷ 2 = 1.5 → 1.5 × 9 = 13.5. Therefore 13.5 L are lost over 9 weeks.
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Extended Examples
1.
Evariste and Gauss have challenged each other to a duel - Nerf guns at dawn! In this duel, they stand back to
back facing away from each other, and then walk away from each other for 10 seconds. After 10 seconds, they
will turn and fire.
Evariste walks at 6 ft/s; Gauss walks at 7 ft/s. How far apart will they be when they fire?
Setup ratio: Don’t have to - unit cost is given!
Evariste walks 6 ft : 1 s, so the unit cost is 6 ÷ 1 = 6. In 10 seconds, he walks 6 × 10 = 60 ft. Gauss walks 7 ft/s,
so in 10 seconds, he walks 7 × 10 = 70 ft. These are in opposite directions, so in total, they are 60 + 70 = 130 ft
apart
2.
Ferris is late for school - again. As he leaves his house, he sees the bus pass right by him. If the bus is going 17
m/s, and Ferris (a track star) runs at 7.5 m/s, how far apart will they be after 15 s?
Method 1 - Compare distances
How far does Ferris travel?: Multiply unit cost (7.5 metres every 1 second) by the number of seconds
7.5 × 15 = 112.5 m
How far does bus travel?: Multiply unit cost (17 metres every 1 second) by the number of seconds 17×15 = 255
m
Therefore, they are 255 − 112.5 = 142.5 m apart after 15 s.
Method 2 - Separation Rate
At what rate are they separating? The bus goes 17 m/s, Ferris goes 7.5 m/s, so the distance between them
grows at 17 − 7.5 = 9.5 m/s. This is the separation rate - the distance between them grows at 9.5 metres a
second.
Multiply separation rate by total time: 9.5 × 15 = 142.5 m/s.
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3.
Winter is coming... and driveways need to be shovelled. Ned knows he can finish shovelling a driveway in 4
hours by himself. Jaime knows he can finish shovelling it in 8 hours. Robert can finish shovelling it in 5 hours.
If they all work together, how long will it take them to finish shovelling one driveway?
Key Point - find how much driveway each shovels in one hour
Ned: 1 driveway - 4 hours → 1 ÷ 4 = 0.25 driveways in 1 hour
Jaime: 1 driveway - 8 hours → 1 ÷ 8 = 0.125 driveways in 1 hour
Robert: 1 driveway - 5 hours → 1 ÷ 5 = 0.2 driveways in 1 hour
Then working together, they finish 0.25+0.125+0.2 = 0.575 driveways in 1 hour.
How long does it take them to finish 1 driveway?
0.575 driveways in 1 hour... How long does it take them to finish 1 driveway?
Setup ratio: 1 hour : 0.575 driveways
Find unit cost: 1 ÷ 0.575 ≈ 1.739 ≈ 1.74. Therefore, the unit cost is 1.74
hours for 1 driveway.
Conclude: It takes them 1.74 hours to finish the driveway, if working together.
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Problems!
As before, questions marked with one or more “*” are considered more challenging
and require you to think a little more.
Solve these rate/ratio problems.
1. Tyson needs 5 cages to hold 12 carrier pigeons. If he has 48 carrier pigeons, how
many cages does he need?
2. Muhammad Ali can run 5 miles in 40 minutes. Assuming he doesn’t change his
pace, how long does it take him to run 15 miles?
3. If the pipes in the basement leak 3 litres of water every 2 weeks, how much is
lost over 6 months? Assume each month has 4 weeks.
4. Al and Bugsy are selling jeans. Al offers a deal - 5 jeans for $130; Bugsy offers 2
jeans for $50. From a strictly mathematical point of view, who offers the better
deal? (Hint: Compare unit cost for one pair of jeans).
5. If 100 soldiers can eat 100 dozen eggs in a week, how long will it take 400 soldiers
to eat 400 dozen eggs?
6. Ronny takes 5 weeks to paint 6 houses. Dylan takes 4 weeks to paint 5 houses.
Who will finish painting 120 houses first, assuming they maintain the same rate
and start at the same time?
7. The ratio of girls to boys in a class is 6 : 4. If the class has 30 students in total,
how many girls and how many boys are there?
8. Dale drives east at 120 km/h; Jeff drives west at 150 km/h. If they leave at the
same time, how far apart are they after half an hour?
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9. Achilles and Hector are in a race. Achilles runs at 15 m/s, Hector at 13 m/s.
(a) How far apart will they be after 20 seconds?
(b) How long does it take them to become 160 m apart?
10. Nancy can plant 50 flowers in 30 minutes. When she works together with Sid, they can plant
125 flowers in an hour. If Sid was working alone, how long would it take him to plant 50 flowers?
11. Six wolves can eat six sheep in six days. How many wolves does it take to eat sixty sheep in
sixty days? How many days does it take three wolves to eat three sheep?
12. * A high school track is 400 m long, with two straight away sections of 100 m and two semi-circle
sections of 100 m, like below. Will runs at 7 m/s. Colin runs at 6 m/s.
100 m
100 m
(a) If they both start at the same time and maintain the same pace, which section of the track
(there are four in total) will they be in after 4 minutes?
13. John, Cam, and Tara are painting the living room. Jon can paint a living room in 8 hours; Cam
takes 6 hours; Tara takes 4 hours. How long does it take them to paint the living room if they
work together?
14. Dale drives north at 120 km/h; Jeff drives west at 50 km/h. If they leave at the same time,
how apart are they after half an hour? (It’s not 85 km - hint: draw a picture and look up the
Pythagorean Theorem)
15. * The Three Stooges are having a pie eating contest. In 3 hours, Moe can eat 36 pies, Larry
can eat 30, and Curly can eat 60. How many hours does it take them to eat 126 pies?
The following are optional challenge problems.
1. ** One and a half hens can lay one and half eggs in one and a half days. How many eggs does
one hen lay in one day? (Your answer does not have to be a whole number)
2. *** 13 men and 7 boys can finish a job in 7 days, while 6 boys and 13 women can finish the
same job in 6 days. In how many days can 1 man, 1 boy and 1 woman working together finish
the same job (assume constant rates each for men, women, and boys)?
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